The report claims that the Black Eagles are considering their options with Gokhan Tore set to leave for West Ham.
Shaqiri would be brought in as a replacement although the Istanbul giants have already started planning for Tore’s departure since last January after Aras Ozbiliz was signed from Spartak Moscow.
The Switzerland international would be a like for like swap with Tore if the transfer were to go ahead but he does still have another four-years remaining on his contract.
Shaqiri joined Stoke on a £12.75m move from Inter Milan last summer scoring three times and providing six assists in his first season in the Premier League.
Besiktas won the league title last season and would be able to offer Shaqiri Champions League football, something that could be able to tempt the 24-year-old winger.